Related papers: Hausdorff-Young type inequalities for vector-value…

We work in a discrete model of the nonlinear Fourier transform (following the terminology of Tao and Thiele), which appears in the study of orthogonal polynomials on the unit circle. The corresponding nonlinear variant of the…

The main aim of this paper is to obtain Bochkarev-type inequalities for the anisotropic grand Lorentz spaces. In the classical setting, Bochkarev obtained inequalities of the Hardy--Littlewood type, which reveal the connection between the…

In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property.…

This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a…

Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach…

We establish new optimal reversed Hardy-type inequalities on the cone of decreasing sequences from $\ell^p$-spaces with power weights, as well as estimates between different norms in Lorentz spaces of sequences. Based on these inequalities,…

The aim of this work is to establish some cases of the Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group for the fractional Sobolev spaces. Here we work with the fractional Sobolev spaces as given by Adimurthi and Mallick in…

Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $\Delta^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that,…

We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. As a by-product, we provide a Montel-type theorem for the Hardy space of Dirichlet series. This approach also gives an…

Let $H$ be a Hilbert space that can be embedded as a dense subspace of a Banach space $X$ such that the norm of the embedding is equal to $1$. We consider the following statements for a nonzero vector $\varphi$ in $H$: (A) $\|\varphi\|_X =…

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions…

We show a Dvoretsky-Rogers type Theorem for the adapted version of the $q$-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the…

Let $2\leq p<\infty$ and $X$ be a complex infinite-dimensional Banach space. It is proved that if $X$ is $p$-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space…

We consider a family of jointly Gaussian random vectors $\xi_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(\xi_1)|,\dots,|T_{z_n}…

In this article, we establish three fundamental Fourier inequalities: the Hausdorff-Young inequality, the Paley inequality, and the Hausdorff-Young-Paley inequality for $(l, n)$-type functions on $\mathrm{SL}(2,\mathbb{R})$. Utilizing these…

Recently an operator space version of type and cotype, namely type $(p,H)$ and cotype $(q,H)$ of operator spaces for $1\leq p \leq 2\leq q \leq \infty$ and a subquadratic and homogeneous Hilbetian operator space $H$ were introduced and…

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq…

Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality,…

The classical Hausdorff-Young inequality for locally compact abelian groups states that, for $1\le p\le 2$, the $L^p$-norm of a function dominates the $L^q$-norm of its Fourier transform, where $1/p+1/q=1$. By using the theory of…

We present some open problems and describe briefly some possible research directions in the emerging theory of Hardy spaces of Dirichlet series and their intimate counterparts, Hardy spaces on the infinite-dimensional torus. Links to number…